Sensors for food industry: two cases of study
Coordinatore:
Chiar.mo Prof. Marco Locatelli
Tutor:
Chiar.mo Prof. Giovanni Chiorboli
Dottorando: Cristian Tagliaferri
Gennaio 2012
1.1 Impedance Spectroscopy . . . 5
1.1.1 Introduction to IS . . . 5
1.1.2 Metal-Electrolyte interface . . . 6
1.1.3 State of the art . . . 12
1.2 Sensor analysis . . . 14
1.2.1 Analytical study . . . 16
1.2.2 FEM simulations . . . 20
1.3 Measurement bench . . . 23
1.3.1 Instruments and methods . . . 23
1.3.2 System block diagram and initial connections setup . . 25
1.3.3 Preliminary tests and improvements . . . 27
1.4 Measurement results . . . 30
1.4.1 DC measurements . . . 32
1.4.2 Water conductivity . . . 33
1.4.3 Conductivity of standard solutions - OIML R56 . . . 36
1.4.4 Measurements on vaccine milk . . . 36
1.4.4.1 Temperature influence . . . 39
1.4.4.2 Fat content . . . 40
1.4.4.3 Milk sophistication with water addition . . . . 40
1.4.4.4 Milk contamination with ammonia . . . 44 1.4.4.5 Milk contamination with benzalkonium chloride 45
2 Variable Electrochemical Cells with Parallel Plate Sensors 47
2.1 First prototype . . . 50
2.1.1 Cell calibration . . . 51
2.1.2 Preliminary measurements results . . . 55
2.2 Second prototype . . . 57
2.2.1 Calibration . . . 58
2.2.2 Measurement results . . . 58
3 Ultrasound Velocity Sensor 63 3.1 Basic ultrasounds theory . . . 63
3.2 State of the art . . . 66
3.3 System description . . . 68
3.3.1 Measurement cell . . . 69
3.3.2 Transmitting circuit . . . 70
3.3.3 Receiving circuit . . . 75
3.4 Digital Signal Processing . . . 76
3.4.1 Zero-crossing estimation . . . 78
3.4.2 Sampling frequency effect . . . 79
3.4.3 Influence of the interpolation-filter bandwidth . . . 79
3.4.4 Influence of the number of bit . . . 80
3.4.5 Noise Effect . . . 82
3.4.6 Zeros-crossing times following the first ones . . . 82
3.5 Conclusions . . . 83
4 Fringe activity: Pulsed periodic flow rate monitoring using a hot-film anemometer [42] 85 4.1 Introduction to the problem . . . 85
4.2 The designed system . . . 86
A LCRcontroller Overview 95
Bibliography 99
Acknowledgements 105
on the whole. . . 9 1.4 More realistic schematization of metal-electrolyte interface. . . 10 1.5 Energy and potential levels through the metal-electrolyte inter-
face in equilibrium state (dashed line) and excited state (full line). . . 11 1.6 Butler-Voltmer equation for the current flowing through a metal-
electrolyte interface. . . 13 1.7 Interdigitated sensor. . . 15 1.8 Transformation of the section of a 2-finger interdigitated sensor
using conformal mapping. . . 17 1.9 Cross-section view of electric field lines of a IDAM’s FEM simu-
lation (simplified geometry). Lower part is made of borosilicate glass while in the upper part r = 1 is imposed. It should be noticed that the electric field is not affected by different permit- tivity values. On the right, the electrical field intensity along the dashed line is shown. . . 21 1.10 The measurement method. . . 23 1.11 Solartron impedance accuracy chart. . . 25 1.12 Block diagram of the measurement system. For the connection
between the computer and the switching unit two kind of inter- faces are possible depending on the switch used (see the next subsection). . . 26 1.13 The connector used for electrically connect the sensor. . . 27
1.14 Initial set-up of the switching unit based on the Agilent 34970A Data Acquisition/Switch Unit. . . 28 1.15 Final connection setup of the switching unit based on the de-
signed MuxBoard. . . 29 1.16 50 Ω resistor measurement results. Black line refers to 1 m long
cables. Grey line refers to 10 cm long cables. . . 30 1.17 Equivalent circuit of IDAM in air. . . 31 1.18 I-V measurements on milk. After each voltage change, a short
delay is necessary before measuring the current for settling. . . 33 1.19 Modulus and phase of the admittance for several amplitudes of
the excitation voltage applied to a sensor dipped into a milk sample. The reported amplitude should be intended as rms. . . 34 1.20 Real part of the admittance measured for three different sam-
ples of water: distilled water from the clean room of the In- formation Engineering Department (blue), distilled water from the Department of Chemistry (green), and drinking water (red).
Each measurement has been repeated two times. . . 34 1.21 Conductivity of several KCl solutions at 1 MHz compared with
the values reported in the standard [33]. . . 37 1.22 Equivalent circuits of the milk impedance useful for the complex
interpolation of the data. . . 38 1.23 Admittance at 1 MHz as a function of temperature. Both exper-
imental data (marker) and linear interpolation (line) are shown. 39 1.24 Conductance from about 200 kHz to 1 MHz for three differ-
ent kinds of milk: skimmed (circles), semi-skimmed (stars) and whole or full-fat milk (squares). Two different series of data are shown, one at 21◦C, obtained after 4 hours, and the other at 4◦C, measured after 19 hours. . . 41 1.25 Admittance as a function of the fraction of water added to UHT
milk at 23◦C. In (b) some measurement repetitions are also reported. Lines refer to polynomial interpolations of the 3rd order for the full range of adulteration (a) and of the 1st order for small fractions of adulterant (b). . . 41 1.26 The modification of the elements of the equivalent circuit with
the water addition. . . 42
1.29 Permittivity and conductivity measured on four different sam- ples as a function of frequency. In the four samples 1000 ppm (red), 519 ppm (blue), 260 ppm (green) and 130 ppm (light blue) of benzalkonium chloride were added to the distilled water. . . 46 2.1 Parallel plates sensor representation. . . 48 2.2 The first prototype. The colours indicate the material used:
brass for the orange parts and white for the not conductive ones. The plate on the left in the figures is the fixed one while the movable one, on the right, is reported without the external sleeve of Teflon. . . 50 2.3 The device complete of the mechanical handling. . . 52 2.4 Experimental results in air (squares) compared with the ex-
pected values (solid line). . . 53 2.5 FEM simulation of the edge effect. . . 54 2.6 Modified electrode and measurement results. . . 54 2.7 Reactance values as a function of the distance d when distilled
water was inserted in the cell. . . 55 2.8 The set up during a measurement. The red arrows indicate the
direction of the shear stress applied to the mechanical support. 56 2.9 The PCBs used as fixed (on the left) and movable (on the right)
plate of a parallel plate configuration. The rear view is reported for showing the electrical connection with the SMA connector. 57 2.10 Complete sensor configuration and correct positioning during a
measurement. . . 58 2.11 Calibration measurements to be compared with those obtained
with the first prototype. . . 59 2.12 Conductivity and permittivity of distilled water, according to
Eqs. 2.7 and 2.8. . . 60
2.13 Results obtained with milk adulterated by water addition. Pro- cedure A is based on a linear interpolation with d on a single frequency measurements. Procedure B is based on the equiva- lent circuit model, which has been obtained from measurements from 20 Hz to 1 MHz. . . 61 3.1 Matching technique between materials with different acoustic
impedances. . . 65 3.2 Different kinds of ultrasonic sensor systems: velocity meter of
particles using Doppler effect (1a), distance meter using phase shift (1b), material characteristics detector using amplitude at- tenuation (1c) and (2), process monitor sensing ultrasound vi- brations (3), element specific detector using resonances (4). See [36]. . . 67 3.3 TOF estimation using different techniques for the arrival time
detection: first arrival-time (1), threshold-based time (2), and first zero-crossing time (3). . . 68 3.4 Block diagram of the first measurement system for the ultra-
sound velocity. . . 69 3.5 Impedance modulus of the piezoceramic transducer PZT-4E,
2.5 MHz centre frequency, by Medel Italiana. . . 70 3.6 (a) Variation of the peak-to-peak output signal as a function of
the width of the rectangular pulse on the transmitter. Markers refer to experimental results, line to simulations. (b) Single re- sponses to the rising and falling edges of the transmitted pulse, and their combined effect. . . 72 3.7 Measured peak-to-peak amplitude of the received signal for dif-
ferent slopes of the step signal used as stimulus. . . 73 3.8 General outline of the transmitter. . . 73 3.9 Block diagram of the voltage multiplier used for the power sup-
ply of the half bridge in the transmitter. . . 74 3.10 Block diagram of the receiver circuit. . . 76 3.11 Block diagram of the signal processing sequence. . . 77 3.12 Standard deviation of the time estimation for different meth-
ods for the zero-crossing detection. Fixed parameters: α = 0.5, fs= 20 MHz, Nbit= 10. . . 79
tions. Fixed parameters: α = 0.5, fs = 15 MHz, and a linear interpolation for the zero-crossing detection. . . 81 3.16 Standard deviation of the time estimation for different SNR.
Fixed parameters: fs = 15 MHz, Nbit = 12, and a linear inter- polation for the zero-crossing detection. . . 82 3.17 Standard deviation of the time estimation for different zeros.
Fixed parameters: α = 0.5, fs = 15 MHz, Nbit = 12, and a linear interpolation for the zero-crossing detection. . . 84 4.1 Test setup of the system. On the left the system controller,
in the centre the mechanical dispenser with the designed flow monitoring board, and on the right the velocity calibrator. . . . 88 4.2 Output of the CTA circuit for different liquids and different
rates: in (a) the dosing period Tdose is 0.8 s while in (b) is 2 s.
Dispensing time tdose is fixed to 0.5 s for both. Trigger signal shows the reference signal provided by the dosing machine and has been scaled down in amplitude for make easier the comparison. 89 4.3 Block diagram of the designed conditioning circuit. . . 90 4.4 CTA signal, filtered signal and conditioning circuit output are
shown. In this measurement a too long impulse for the SSG circuit has been chosen and, as a consequence, the first dose is lost by the monitoring circuit. . . 92 4.5 Trigger signal from the dosing system and final monitor signal
provided to the system controller. . . 92 4.6 Simulation of the proposed alternative circuit for simplify the
comparison made by the system controller to a simple threshold crossing. . . 93 A.1 Main screen of the LCRcontroller. . . 96 A.2 Flowchart of the automatic range selection feature with the im-
plemented solution for manage the unjustified overflow indication. 97
A.3 Flowchart of the added system for limit the autorange selection iterations. . . 98
amounts of mesh elements. Only the overhanging air has been simulated, the results have been multiplied for air + glass to estimate the total capacitance. . . 22 1.3 Simulation results with different numbers of fingers but using a
similar number of degrees of freedom normalized to the amount of fingers. . . 22 1.4 Nominal and measured conductivity of different kinds of water. 36 1.5 Main constituents of cow milk with average percentages. [34] . 37 1.6 Fat content of different kinds of milk for 100 ml. . . 40 1.7 Interpolation parameters A, B and C in equations (1.32), (1.33)
and (1.34). ρ represents the regression coefficient. . . 43 2.1 Comparison between expected and measured values. . . 53 2.2 Comparison between nominal and measured values for some
test liquids. . . 60 2.3 Comparison between variable electrochemical cell (PPS) and
IDAM sensors. Different procedures have been used to obtain the reported results: linear interpolations on several measures at 1 MHz with different gaps (Procedure A) and complex inter- polations on the whole frequency range (Procedure B and Interp). 62 4.1 Distilled water and ethanol parameters. . . 88
The research has concerned liquids analysis for food industry through two main techniques: impedance spectroscopy and ultrasounds. The first one has been the primary subject for the first part of the course, while the latter has been developed in the last year. In addition, during these years, a pulsed flow monitoring system (not for the food industry) has been designed as a fringe activity.
Impedance spectroscopy has been applied to the quality control for the dairy industry in order to support classical laboratory analyses by continuous, in- line, measurements. Direct reports, from the production chain, of possible abnormalities in the electrical parameters will increase the efficiency of the system and the quality of the final product. In particular, the sensitivity to some milk adulterants of not functionalized InterDigitated Array Microelec- trodes sensors, IDAMs, has been analysed.
Functionalized IDAMs have been successfully used in recent years for de- tecting pathogenic bacteria, or contaminants, since they are able to emphasize the contribution of materials on their surface. If not functionalized IDAMs could detect other common adulterants, like water; an array of similar sensors (functionalized or not) might be used for a more complete characterization of beverages.
Initially, the effect of the sensor’s geometry on the electric field has been investigated for understanding the thickness of the sensitive layer over the
electrodes; then several measurements of different solutions (milk and water, milk and ammonia, etc. ) in different environmental conditions have been performed.
Since a limiting condition for the IDAMs use was found in the measure- ments of high conductive solutions, a variable electrochemical cell, based on the classical parallel electrode configuration, has been designed. The distinc- tive feature of this new sensor was the adjustable gap between the electrodes.
Two different prototypes were realized because of some sealing problems of the first one. This cell has been able to overcome the limitation of the IDAMs at the cost of a more complex measurement procedure.
The second technique developed, employing an ultrasonic sensor, concerns the continuous in-line measurement of substances concentration or density since it allows improving some industrial processes, for instance the primary fermentation of wine or beer.
The system is a time-of-flight (TOF) meter, which measures the ultrasonic velocity in the liquid, because density depends on the propagation speed of such pressure waves.
The whole system has been designed to transmit a front wave and read the pulsed oscillation on the receiver. A new technique has been studied for estimating the TOF and the design of the full-digital receiver depends on the results of several simulations. The complete system, with the receiver and the transmitter, has been implemented and characterized in its singular compo- nents, even if experimental results of the complete system are not currently available.
Finally, the fringe activity concerns the design and the verification of a flow monitoring system which must verify the correct functioning of a glue dispenser used in an industrial production process. The monitored flow has a pulsed be- haviour with period of the order of seconds, therefore standard techniques cannot be used to compensate the sensor characteristics. The uncompensated output, coming from an anemometer used in constant temperature configura- tion, has been filtered to obtain a signal that will be compared with a reference waveform by an external microcontroller. The main issue of such an approach is the long settling time because of the low frequency poles of the filter. To
a fixed threshold. This allows to verify the dispensing of a minimum defined quantity of glue and detect possible blocks.
1.1 Impedance Spectroscopy
1.1.1 Introduction to IS
Impedance spectroscopy, IS, is a technique for gathering physicochemical in- formation of a substance from its response to a small signal stimulus. Analysis is generally carried out in the frequency domain and numerical data obtained from measurements are post-processed in order to point out more information compared to the simple impedance evaluation.
Classical procedures consist in applying a low voltage signal, generally less than 1 V to avoid electrochemical reactions as will be later explained, and measuring the magnitude and the phase of the flowing current. Depending on the electrical properties of the tested material, very large ranges of values are possible so that it is commonly more useful to express the measured values in two different ways:
• Impedance (Z), i.e. the ratio of the applied voltage over the measured current, for low impedance values.
• Admittance (Y), i.e. the ratio of the measured current over the applied voltage (the inverse of impedance), for high impedance values.
Impedance and admittance, in turn, can be written as:
Z = R + jX , (1.1)
Y = G + jB , (1.2)
where the resistance, R, and the conductance, G, are the real parts while the reactance, X, and the susceptance, B, are the imaginary parts of the impedance and the admittance, respectively.
When an insulator, or a high impedance material, is measured, another numerical representation is normally used by means of the permittivity ():
Y = jωrCvacuum = jω0rCvacuum+ ω00rCvacuum, (1.3) where r = 0r− j00r is the complex relative permittivity and Cvacuum is the capacitance of the sensor in vacuum. The product ω00r0, where 0 is the vac- uum permittivity equal to 8.85 × 10−12F/m, is better known as conductivity (σ). The use of a complex permittivity allows to enclose the real part of the admittance, the losses, in the expression.
In impedance spectroscopy very different kind of sensors can be used, so different geometries are possible. The main problem in these situations is to find a way to compare measurements results because different geometrical dimensions lead to different absolute values of impedance. For this reason it is helpful to introduce the cell constant, K in m−1, that takes into account the physical dimension contribution only. Using this parameter, distinctive of each sensor, the measured resistance and capacitance become:
Rmeas= ρM U T · K , (1.4)
Cmeas= 0rM U T
K , (1.5)
where ρM U T and rM U T are the resistivity (the inverse of the conductivity σ) and the relative permittivity of the Material Under Test, MUT, respectively.
From Eqs. 1.4 and 1.5 it is so possible to obtain the physical parameters of the material regardless of the geometry of the sensor.
1.1.2 Metal-Electrolyte interface
When a metallic conductor is dipped in an electrolyte, i.e. a solution in which molecules dissociate in ions, the metal atoms directly touched by the liquid
(a) Adsorption of superficial metal ions. (b) Adsorption with polar liquids.
Figure 1.1:Adsorption effect at the metal-electrolyte interface.
tend to ionize losing an electron. These electrons move inside the bar so that the bulk remains overall neutral.
The ionized metal atoms of the bar are electrostatically attracted to the metal surface but they cannot penetrate it as shown in Fig. 1.1(a). Such phe- nomenon is called the adsorption of the metal ions onto the surface of the object. The same effect is obtained using liquids made up of polar molecules in which atomic bonds make that different parts of the same molecule have different electrostatic charges. As an example, water molecules are polar be- cause of their banana-shape and then it is possible to have the adsorbition effect (Fig. 1.1(b)).
The most important parameter for quantify such an effect is the reactivity of the metal: reactive metals ionize very easily because of their weak electrons attraction. Sodium and zinc atoms ionize easily oppositely to gold or platinum that hardly ionize.
Ionization depends on several forces that play different roles on the final balance:
• polar molecules encourage metal surface ionization;
• recombination in the metal reduces ions amount;
• negative voltage applied to the bar produces electrons excess that limits metal ionization;
Figure 1.2: Double layer in a metal-electrolyte interface.
• positive voltage applied to the bar drains out electrons allowing new metal atoms to ionize.
Referring to Fig. 1.2, the metal ions adsorbed onto the plate form the Adsorbed Fixed Layeras well some negative charged ions in the electrolyte are attracted to the same metal ion layer forming a Diffuse Mobile Layer.
These two layers together are called Electrical Double Layer and their behaviour is quite similar to the capacitor’s one.
As above mentioned, applying a positive voltage to the metal plate, some electrons are drained out leaving some metal ions free to move into the elec- trolyte. The final result is that the neutral charge of the bar is restored. At this point other atoms are able to ionize according to the metal reactivity and so new free electrons are generated and ready to be drained out from the battery. Therefore this process allows a constant current flow (consuming the electrode) similar, in the electronic circuit theory, to a resistive behaviour . This resistance is named Faradaic Resistance.
Using an alternate voltage generator instead of the continuous voltage one, a different effect is obtained because no corrosion effects are involved when no net flow of current is present. The diffuse mobile layer, made up of negative ions, is placed to a short distance from the bar proportional to the charge of the free electrons, negative as well, in the metal. Removing free electrons reduces the electrostatic repulsion with negative ions in the electrolyte so the mobile layer is able to move closer to the metal surface. In an opposite way, adding electrons extend the influence area of the negative charge in the metal
Figure 1.3: Capacitive behaviour of the diffuse mobile layer and the adsorbed fixed layer on the whole.
so the mobile layer is forced to move away from the plate. Adding or removing negative charge is equivalent to reduce or, respectively, increase the electric potential and, as illustrated in Fig. 1.3, current is 90◦ earlier than the potential showing a typical capacitive behaviour. Such a behaviour is modelled using the well known Double Layer Capacitor.
Actually the real situation at metal-electrolyte interface is more complex and less defined. In Fig. 1.4 are reported several layers that better describe the system:
• Inner Helmholtz Plane (ihp) referred to the metal ions adsorbed onto the metal electrode;
• Outer Helmholtz Plane (ohp) is the closest distance to the electrode for the free solvated electrolyte ions;
• Diffuse Region of Charge is the area in which electrolyte ions can still influence the interface behaviour. This region has a characteristic Debye length, which is typically on the order of 10 Å;
• Diffusion Layer is the last zone in which the interface affects the free ions in the electrolyte; the ions concentration quickly return to neutrality moving away in the bulk;
Figure 1.4: More realistic schematization of metal-electrolyte interface.
• Bulk Solution is not affected by influences and ions are randomly dis- tributed.
Only the first three layers (ihp, ohp, and the diffuse region of charge) are considered to be part of the interface.
Having a closer look at the electrochemical system subjected to an external potential difference, it is possible to write:
V(I) = E + η + R · I , (1.6)
where V(I) is the potential applied to the electrodes (the metal plate and the electrolyte), E the internal potential in equilibrium state (without any external stimulus applied), R is the electrolyte resistance, I the current in the system and η is the overvoltage element that include every unbalanced phenomenon like the activation overvoltage and the concentration overvoltage (diffusion, reaction and crystallization).
The activation overvoltage is the potential used for the charge transfer (ionization and recombination) and it represents a typical threshold reaction so that only the atoms with higher energy than the activation threshold are able to react.
According to the polarity of the applied potential, two opposite reactions are possible:
• Anodic reaction M −→ M++ e− (oxidation)
cathodic polarization, from Fig. 1.5 it is easy to gather that:
WAT T,A= WAT T,AEQ + zF η(1 − α) − zF η = WAT T,AEQ − αzF η , (1.7) WAT T,C = WAT T,CEQ + zF η(1 − α) , (1.8) where z is the number of free electrons generated in the ionization process and F is the Faraday Constant equal to 96 485 C/mol.
Figure 1.5:Energy and potential levels through the metal-electrolyte interface in equilibrium state (dashed line) and excited state (full line).
Using Arrhenius equation for the reaction rate, anodic and cathodic density currents can be written as:
jA= zF kACrede−WAT T,ART = zF kACrede−
WEQ
AT T ,A−αzF η
RT , (1.9)
jC = zF kCCoxie−WAT T,CRT = zF kCCoxie−
WEQ
AT T ,C+(1−α)zF η
RT , (1.10)
where k is the kinetic constant of the reaction, C is the concentration of the reagents (reducing or oxidizing) and R is the ideal gas constant equal to 8.314 J/(K mol).
Without external excitation, at equilibrium ( η = 0):
jA= −jC = j0= zF kACrede−
WEQ AT T ,A
RT . (1.11)
Finally, adding both the contributions, the total current is:
j= jA+ jC = j0(eαzF ηRT − e−(1−α)zF ηRT ) . (1.12) Eq. 1.12 is known as the Butler-Volmer equation and is graphically re- ported in Fig. 1.6.
It is easy to notice that the electrical response of a metal-electrolyte inter- face is similar to an antiparallel diodes configuration’s behaviour, so, in order to remain in quite linear conditions, signal amplitude used for the measure- ments has to be carefully chosen.
1.1.3 State of the art
The classical methods used in physiological analysis are based on chemical or biological techniques. These procedures take from several hours to some days so their in-line application is not possible and new ways have been sought.
Impedance measurements have been recently used for bacteria detection and quantification using special shaped sensors or bio-recognition elements on the surface of interdigitated electrodes [4, 5]. For example making up a planar microchannel so that cells are able to pass one by one, allows an ac- curate counting for low concentrations by measuring the impedance along the channel section. For higher concentrations, or growth monitoring, the inter- digitated configuration is also used with success just reading the impedance of the bacteria layer that covers the fingers of the sensor [6, 7].
Figure 1.6: Butler-Voltmer equation for the current flowing through a metal- electrolyte interface.
Coupled with bio-recognition elements, which react with specific cells or molecule, the interdigitated electrodes are frequently used because of their high sensitivity close to their surface. When the target molecule reaches the selective material that cover the sensor, the bio-recognition element reacts changing its own impedance allowing an easy detection [4, 5, 8, 9, 10, 11].
With interdigitated sensors indirect measurements for cells growth control are also possible; instead of directly quantify the number of organisms, metabolites produced by bacterial cells, or even the charge of their internal components, can be monitored [5].
In dairy industry adulteration tests are a very common problem. Water addition slightly changes milk properties and this makes the detection hard.
Cryogenic verification is the standard procedure because milk freezing point, normally occurred between −0.525◦C and −0.530◦C, depends on water per- centage. The greater the amount of water, the higher the freezing point. The main problem in milk analysis is the variability of its composition, since sea- son, breed cattle, cows diet and thermal processes can affect the freezing point.
For that reason −0.520◦C is generally used as acceptable threshold.
Another remarkable problem is the addition of cooking salt to the milk used to mask the adulteration. As it is well known, salt is able to reduce the freezing point so the simultaneous addition of the correct water and salt combination can result in a freezing point similar to that of pure milk. Nowadays chloride analysis is used, together with cryogenic measurement, in order to check for adulteration of milk.
A further technique for determining the freezing point of milk is based on the Fourier transform infrared technology and some devices based on that principle are already commercially available [12].
Focusing on impedance measurements, their applicability in milk adulter- ation detection has already been verified using simple planar sensors [13, 14, 15, 16] or standard conductivity meters [17] showing a good sensitivity when ultra pure water is used. It is not clear the relation that connects the percent- age of water with the conductivity of the milk because [15] reports a linear trend while [17] a power-law one. [16] also reports an unusual behaviour for 1-2 % of added water that suddenly reduces the conductivity of the milk to be later increased again for higher water concentrations.
Others conductivity measurements on the milk have been performed using commercial conductivity cells, or with couples of electrodes, to verify fat or cream content [18], synthetic milk [19], and acidification [20].
1.2 Sensor analysis
InterDigitated Array Microelectrodes, IDAMs, are planar sensors made of an insulating substrate on which a thin conductive layer is deposed and moulded to form an interdigitated configuration, see Fig. 1.7(a).
Impedance measurements with this kind of sensors are commonly used for specific bacteria detection covering the surface of the electrodes with bio- recognition elements [4, 5, 8, 9, 10, 11]. These films are able to bind to them- selves only the bacteria they are optimised for. As it will be better showed in the next sections, interdigitated sensors are able to analyse only a thin layer of solution next to their surface. The thickness of this layer directly depends on the gap between two sided fingers of the interdigitated sensor, so geometric pa- rameters are crucial for each specific application: narrow gaps ensure no bulk influence whereas wide spaces allow thick solution layers to be involved. The greatest benefit of using these sensors in biological analysis is the remarkable time reduction, due to their ease of use, compared to the classical laboratory
(a) Interdigitated geometry. (b) Picture of the ABTECH sensor used.
Figure 1.7:Interdigitated sensor.
methods that should be utilized for more detailed investigations anyway.
The same method can be exploited more times on the same substrate mak- ing a so called Lab-on-a-Chip, LOC, in which different interdigitated electrodes analyse the same solution at the same time but looking for different kinds of bacteria.
IDAMs are also used for bacteria counting and estimation when it is im- portant to know the growing level in a culture [6, 7]. As above mentioned, the size of the fingers and the size of the bacteria influence the accuracy of the estimation.
This kind of sensor has been chosen, for our measurements, with the pur- pose of verifying its sensitivity in milk quality control and comparing the obtained results with other kind of sensors. The final target of this investiga- tion is to explore the possibility of using IDAMs in the milk production and distribution chains, for a real on-line monitoring.
The IDAMs sensors used in the measurements are manufactured by the ABTECH Scientific [21]. In Fig. 1.7(b) a picture of a IME 2050.5 M- Au-U sample is shown. Insulating substrate is made of 0.5 mm thick Schott D263 Borosilicate Glass (relative permittivity of 6.7 @ 1 MHz and resistivity of 1.6 × 108Ω·cm) 2.00 cm long and 1.00 cm large. A 100 nm gold layer is deposed on the substrate and an interdigitated pattern is obtained using microlitho- graphic techniques. Finally, a silicon nitride protection layer is placed on the sensor only leaving the interdigitated fingers and the bonding pads uncovered.
The sensing area, limited to the interdigitated fingers only, is composed of 50 couples of 4.980 mm long and 20 µm wide fingers. Interdigit spacing between fingers is 20 µm.
For a better understanding of the IDAM behaviour, two different ap- proaches, analytical and numerical, have been followed. The first one allowed to quantify the electric field and the capacitance as a function of the interdig- itated geometry whereas the second one, based on finite elements simulations, has been used as a comparison and check tool.
1.2.1 Analytical study
As already mentioned in the introduction, the key parameter of an impedance sensor is the cell constant which only depends on the geometrical dimensions.
Through this value it is possible to find out the real electrical parameters of a solution, the conductivity σ and the permittivity , from a single measure.
In the technical datasheet of the IDAMs [22] a nominal value of 0.040 cm−1 for the cell constant K is reported, so that the theoretical capacitance in air should result:
C= Cair+ Cglass
= 0
K + 0glass K
= 0
K ·(1 + glass)
= 8.85 × 10−12[F/m]
4 [m−1] ·(1 + 6.7) = 17.04 pF . (1.13) As it is possible to notice in Eq. 1.13, the total capacitance is made of two contributions, one concerning the air over the interdigitated fingers and one ascribed to the glass of the substrate. In air, the resulting capacitance is mainly due to the substrate.
The cell constant can be determined by analysing the electric field around the sensor. In literature several analytical methods have been reported [23, 24, 25, 26, 27, 28], mostly based on the conformal mapping technique.
Formally, a conformal map is a mathematical function which preserves an- gles. This transformation allows to map an harmonic function defined over a particular space, like an electromagnetic field, to another space preserving its harmonicity. In practical terms, it is possible to study the electric field of a complex geometry transforming it in a more convenient one.
(a) Original plane. (b) Transformation to Z-plane.
(c) Transformation to T-plane. (d) Transformation to W-plane.
Figure 1.8: Transformation of the section of a 2-finger interdigitated sensor using conformal mapping.
A deep literature study has been made with the aim of understanding this approach and allowing a more thorough comparison. Only a simplified review and a short discussion about the main distinctive features of several approaches will be reported.
Instead of considering the whole extension of an interdigitated sensor, it is more practical to reduce the analysis to two sided fingers only, so that the issue is limited to the analytical expression of the capacitance between two sided plates. In Fig. 1.8(a) the cross-section of the fingers is shown and only the beneath part will be considered in the estimation, the same equations will be applied to the over part for the symmetry of the geometry. That configuration can be further simplified, Fig. 1.8(b), using other symmetries: in a periodic system only the right half of a finger concurs to the capacitance with the finger on the right (similarly for the left half, with the left finger) and, furthermore, the system can be seen as the sum of a series of two capacitors of doubled value, using a virtual equipotential line along the CC’ symmetry line.
At this point a Schwarz-Christoffel transformation is applied to map the 0346 polygon in the Z-plane to half plane of the T-plane using the mapping function:
t= cosh2πz 2h
. (1.14)
As it is possible to see in Fig. 1.8(c), the whole perimeter of the 0346 polygon in the Z-plane is mapped onto the real axis of the T-plane and the vertices will assume new values:
t0 = 0 , t1 = cosh2πs 2h
, t2 = cosh2π(s + g) 2h
, t3 = t4 = ∞ , t5 = − sinh2π(s + g)
2h
, t6 = 0 .
Finally, the upper half of the T-plane is mapped onto the interior of a rectangle in another plane using again a Schwarz-Christoffel transformation:
w= AZ t5
t
dt
(t − t0)(1−απ)(t − t1)(1−βπ)(t − t2)(1−γπ)(t − t5)(1−δπ) + B
= AZ t5
t
dt
p(t − t0)(t − t1)(t − t2)(t − t5) + B ,
(1.15)
where α, β, γ, δ are the angles of the vertices in the final W-plane, fixed to π/2 in a rectangle, and A and B are constants determined from the coordinates of the final rectangle, Fig. 1.8(d).
Only at this point, in the W-plane, it is possible to easily estimate the capacitance with the classical parallel plates equation. Afterwards, the result should be brought back to the original plane using the inverse functions of the mappings. As reported in [24] the final solution is:
C= 1
2r0K(k)
K(k0), (1.16)
where the 1/2 is due to the series of the two identical capacitance in which the system has been divided, K(x) is the elliptic integral of the first kind, and k and k0 are defined as:
k= sinh πs2h sinhπ(s+g)2h ·
v u u u t
cosh2π(s+g)2h + sinh2π(s+g)2h
cosh2 πs2h+ sinh2π(s+g)2h , (1.17)
k0 =p1 − k2. (1.18)
K= 0r
C , (1.19)
where r
0 is the composition of the permittivities of the several layers.
As just said, for a complete evaluation of a real sensor more detailed cal- culations must be done. A review of different methods is now reported:
1. Olthuis [23] uses a more elaborate transformation for mapping the whole two-finger configuration without simplification due to the symmetries.
A simple multiplication for the number of gaps between the fingers is counted and a different expression for the two-finger configuration is provided.
2. Gevorgian [24] provides a very complete analysis of the interdigitated geometry taking into account:
• the thickness of the fingers, increasing their effective width using Wheeler’s first order approximation;
• the different behaviour of the external fingers, on the basis of results obtained with a three-finger geometry;
• multilayer substrate (or even superstrate) with different permittiv- ity, by adding their individual contributions;
• the capacitance due to the finger ends, obtained by applying again the conformal mapping;
• the two or three-finger particular configurations.
3. Igreja [25] includes in the computation the different effect of the marginal fingers and the multilayer substrate or superstrate.
4. Otter [26] proposes a numerical approximation of the potential that allows reducing the computation to a sum of Bessel functions of the first kind; it refers only to two sided fingers of a larger array without others closer examinations.
Table 1.1: Analytical values of the constant cell for the ABTECH interdigi- tated sensor used in this work. Values with r= 1 over the sensor.
Method Estimate Nominal [22] 4.000 m−1
Olthuis [23] 4.057 m−1 Gevorgian [24] 4.859 m−1 Igreja [25] 4.039 m−1 Otter [26] 4.014 m−1 Zaretsky [27] 4.016 m−1
5. Zaretsky [27], starting from [28] and using fingers width equal to gaps width, shows that the cell constant is equal to 2/(NL) where N is the number of fingers and L their length. This analysis is not based on the conformal mapping but on a direct analysis of the electrical field distribution.
A table that summarize the numerical results of these methods (applied to the used sensor) is reported in Table 1.1. It can be noticed that, as reported in [25], the values computed with the Gevorgian model overestimate the con- stant K. The other models provide a good estimate, particularly those in [27]
or in [26], if the correct number of electrodes is considered.
1.2.2 FEM simulations
Another technique to estimate the capacitance, and consequently the cell con- stant, is the numerical simulation with the finite element method. In this case the geometrical domain is divided in smaller parts on which the Maxwell equa- tions will be solved and the solutions will be agreed with the corresponding ones calculated on the neighbouring subdomains.
Only a cross section of the sensor has been drawn for the simulation, so the resulting capacitance should be multiplied for the length of the facing fingers (5 mm - 20 µm = 4.98 mm) in order to achieve the overall value. The electric field lines are established in the glass substrate as well as in the air above the sensor, so it is necessary to draw these layers with a thickness that include most of the same field lines. As it is possible to see in Fig. 1.9, the electrical field intensity is reduced by half in the first 15 µm from the air/substrate interface
Figure 1.9: Cross-section view of electric field lines of a IDAM’s FEM simu- lation (simplified geometry). Lower part is made of borosilicate glass while in the upper part r = 1 is imposed. It should be noticed that the electric field is not affected by different permittivity values. On the right, the electrical field intensity along the dashed line is shown.
and the main part is located in the first 100 µm. For this reason, 100 µm is the thickness used in the simulations.
As a boundary conditions, a potential of 1 V has been applied to the first finger on the right, grounding the second one and so on alternating the ap- plied potential. The capacitance has been determined adding all the superficial charges built up on the fingers of the same branch and dividing them for the used differential potential (1 V).
A critical parameter for the results of the simulation is the amount of the mesh elements. Small quantities allow faster simulations but with limited ac- curacy, vice-versa higher numbers provide greater accuracy at the cost of long simulation times. In Tab. 1.2 the trend of the resulting capacitance for a whole 50-pair geometry is reported. It has been considered only the overhanging air for the simulation in order to reduce the complexity. For the mesh refinement, the adaptive algorithm has been chosen, so that the simulator made thicker the mesh elements only where necessary instead of uniformly increase them.
Another trick to help the convergence of the results is to set up the simulator for refining the mesh several times more in the y-axis than in the x-axis be- cause of the thin layers present in the geometry. From Tab. 1.2 it is possible to gather that the capacitance tends to convergence as the degrees of freedom
Table 1.2: Simulation results of the IME 50-pair fingers using different amounts of mesh elements. Only the overhanging air has been simulated, the results have been multiplied for air+ glassto estimate the total capacitance.
Degrees of freedom Ctotal [pF] K [m−1]
103 × 103 15.44 4.41
226 × 103 15.99 4.26
489 × 103 16.36 4.16
1.04 × 106 16.57 4.11
2.20 × 106 16.70 4.08
4.60 × 106 16.77 4.06
Table 1.3: Simulation results with different numbers of fingers but using a similar number of degrees of freedom normalized to the amount of fingers.
5 pairs 10 pairs 50 pairs Degrees of Freedom 611 × 103 1.41 × 106 5.81 × 106
Cext_f inger 0.212 pF 0.212 pF 0.215 pF Cint_f inger 0.325 pF 0.320 pF 0.328 pF Ctotal_simulated 1.53 pF 3.00 pF 16.36 pF Ctotal_derived 1.52 pF 3.09 pF 16.29 pF
(that depend on the amount of elements) grow, so, for this complex geometry, it is not worth to exceed the amount of one million.
Instead of use the whole cross section of the sensor, simplified geometries with only 5 or 10 pairs of fingers has been also simulated. In addition to the total capacitance, the individual capacitances of the internal and external fingers have been verified to find out the equation that generalize the overall value with the amount of fingers. The results are reported in Tab. 1.3 and it can be noticed that the external fingers have a different influence than the internal ones so they have to be distinctly handled. The Ctotal_derivedhas been computed using the following equation:
Ctotal_derived = Cint_f inger·(npairs−1) + Cext_f inger, (1.20) that permits to generalize the behaviour regardless the number of fingers. Only
LC
(a) Four-wire connection:
High Potential (HP), High Current (HC), Low Poten- tial (LP) and Low Current (LC).
(b) Auto balancing bridge configuration.
Figure 1.10: The measurement method.
one finger is considered external because at the opposite side the external finger is connected to the other branch and so it should not be taken into account.
1.3 Measurement bench
1.3.1 Instruments and methods
The method used for the impedance measurements is the so called four- terminal sensing[29]. This technique, excluding the parasitic contributions of the connection cables, provides more accurate measures. This connection is shown in Fig. 1.10(a): the ammeter is placed in series with the generator while the voltmeter measures the potential directly across the Device Under Test (DUT).
Instruments for the impedance measurement at low frequencies are com- monly based on the auto balancing bridge circuit. The simplified circuit is reported in Fig. 1.10(b): when an excitation signal (OSC1) is applied to the DUT, the Detector (D) forces another signal generator, OSC2, to modify its output for zeroing the potential on the low terminal of the DUT. When this condition is achieved, the unknown impedance can be estimated as:
Edut
Zx = Err
Rr
→ Zx= Err
EdutRr. (1.21)
The instruments used for the measurements are:
1. Hewlett Packard 4284A Precision LCR Meter (20 Hz to 1 MHz with fixed predetermined steps);
2. Solartron 1260 Impedance / Gain-Phase Analyzer (10 µHz to 32 MHz with resolution down to 10 µHz).
The first one is a simple impedance meter so the arrangement of the four connectors is plain. On the contrary, for the second one it should be paid more attention because six connectors are present on the front panel. The reason for this higher complexity is due to the possibility of using the instrument for measuring the response of a circuit to a sine wave stimulus. For impedance measurements, the GENenerator OUTPUT and the CURRENT input play the role of HC and LC terminals, respectively, of Fig. 1.10(a), while the HI and the LO connectors of Voltage 1 are the HP and the LP terminals.
The accuracy of the instruments is provided from the user manuals in different ways. For the HP 4284A it is possible to calculate the accuracy as:
Eracc= Ae+ Acal, (1.22)
where Ae is the relative accuracy and the Acal is the calibration accuracy reported in a chart at page 9-20 of the HP 4284A operation manual [30]. Ae, in turn, is made of:
Ae= ±[A + (Ka+ Kaa+ Kb+ Kbb+ Kc) · 100 + Kd] · Ke, (1.23) where, for a sample measure at 120 kHz with 100 mV of amplitude on a DUT of (23 + 9i) mS in standard conditions:
• A (Basic Accuracy) = 0.1;
• Ka (Impedance Proportional Factor) = 100 · 10−6;
• Kaa (Cable Length Factor B) = 0;
• Kb (Impedance Proportional Factor) = 204 · 10−6;
• Kbb (Cable Length Factor C) = 1;
• Kc (Calibration Interpolation Factor) = 300 · 10−6;
• Kd (Cable Length Factor E) = 1.75 · 10−3;
• Ke (Temperature Factor) = 1.
Each coefficient has a corresponding formula or table in the operation manual.
Figure 1.11:Solartron impedance accuracy chart.
With the same conditions, Acal = 56 · 10−3, and the total relative accuracy results Eracc'0.2%.
Instead, for the Solartron 1260, the accuracy is reported in a chart and depends only on the frequency and the impedance of the DUT. That chart is shown in Fig. 1.11.
1.3.2 System block diagram and initial connections setup The measurement system is able to sequentially measure three samples using one impedance meter only. The two different instruments allows a comparison of the measures for verifying possible configuration mistakes or instrument- dependent behaviours.
The block diagram of the system is reported in Fig. 1.12. The four cables of the impedance meter (LCR meter) are connected to a switch unit that alternately connects one of the three sensors; these sensors, dipped in the samples, are placed in a climatic chamber and, when an accurate temperature
LCR Meter
Legend
Thermocouple 50 Ω Cables GPIB GPIB or RS232
4 PROBE
1
PROBE 2
PROBE 3 4 4
Switch
4
Temperature Meter
PC
(Matlab GUI)
Climatic chamber
Figure 1.12: Block diagram of the measurement system. For the connection between the computer and the switching unit two kind of interfaces are possible depending on the switch used (see the next subsection).
control is required, the samples are also dipped in a water bath placed in the chamber as well. The temperature of the three samples is measured by means of three thermocouples placed close to each sensor.
The whole system is controlled by a computer through a IEEE-488 GPIB and a RS-232 serial bus. The software has been written in a MATLAB envi- ronment creating a GUI (Graphical User Interface) that allows to completely configure the measurements and even a scheduling for long running time mea- surements (see Appendix A for more information).
For the electrical connection between the four 50 Ω cables and the probes a special connector has been designed, some pictures are reported in Fig. 1.13.
Firstly the four cables, after the switch, are soldered to an Arlon25 PCB (Printed Circuit Board) expressly shaped for connecting the cables in pairs and making the two contact points available. The pads of the sensor are contacted through two elastically deformable thin strips of copper soldered on the PCB, shown in Fig. 1.13(a), mechanically pressed on the sensor itself. Everything is held together using two pieces of Teflon and a plastic screw that allows to adjust the force of the contact. The picture of the complete connector is shown in Fig. 1.13(b).
The initial set-up of the connections is based on the Agilent 34970A
(a) The PCB board shaped for connect the probes.
(b) The whole connector that, pressing the probe against the PCB, provides the electrical contact.
Figure 1.13:The connector used for electrically connect the sensor.
Data Acquisition/Switch Unit with the optional 34901A 20-Channel Multiplexer module. That instrument allows the switching between the three probes changing the states of some relays. Note that the available module does not have coaxial connectors so simple unshielded cables have been used.
In Fig. 1.14 the exact wiring configuration is illustrated, each cable from the impedance meter is switched between the corresponding wires connected to the three sensors.
1.3.3 Preliminary tests and improvements
Once the set-up of the measurement bench was completed, initial measure- ments on the sensor in air have been done. The resulting capacitance at low frequencies was about 190 pF indicating some problems in the system. Several tests in different conditions have been performed and three main issues have been pointed out:
• the unshielded cables, used for the connection of the switch unit, in- terrupt the shielded paths of the signals and the impedance-controlled lines, this adds about 35 pF to the sensor’s capacitance;
• the shieldings connection to earth, carried out through the impedance meters, is interrupted before the switch unit, the coaxial cables connected to the probes are only connected to each others but they are left floating,
Figure 1.14: Initial set-up of the switching unit based on the Agilent 34970A Data Acquisition/Switch Unit.
thus nullifying any effect of the shielding, and increasing the parasitic capacitance of about 70 pF;
• as already said, the general purpose switch unit affects the measurements with additional 70 pF.
Other tests have been done for verifying different configurations for the shieldings of the coaxial cables that, referring to Fig. 1.14, have to be connected in C (see Fig. 1.13(b)) and also in A and in B.
Consequently a new switching board has been designed for this purpose.
On this board four BNC connectors are placed for the impedance meter side while the coaxial cables of the probes are directly soldered next to the relays.
The signal lines are designed in order to have an equal electrical path for each sensor, with a controlled 50 Ω characteristic impedance; in addition the analogic part of the board has been electrically isolated from the digital control part. This switch board, hereafter the MuxBoard, has a RS-232 serial interface using a Modbus protocol, and allows to select one of the three available sensors. The on-board microcontroller just translates the received instructions into the real command for the relays.
For the actual switching DPDT (Double Pole Double Throw) double wind- ing latching relays have been chosen. This entails several advantages:
• there are different reels, set and reset, for connecting or disconnecting each wire;
Figure 1.15: Final connection setup of the switching unit based on the de- signed MuxBoard.
• it is not necessary to keep the reels excited after the commutation, with an appreciable noise reduction;
• the double throw (DT), ensures that the sensors’ cables, when not con- nected to the instruments, are short circuited to ground;
• the same reel operates on two different switches, since they are double pole (DP), thus simplifying the circuit.
The final configuration of the connections is shown in Fig. 1.15 in which it is possible to notice the improved shielding. With this new set-up, the measured capacitance of a probe in air is about 16.7 pF with a cell constant of 4.080 m−1, very close to the theory.
Another issue occurred during the preliminary tests was the reliability of the system at high frequency. When the system based on the Solartron 1260 was used for characterizing a 50 Ω SMD resistor, it shown a resonance above 10 MHz, making the measurements meaningless over that frequency.
Note that for this test the instrument was directly connect to the resistance through coaxial cables without any switches or connectors. This effect can be corrected only reducing the cables length. In Fig. 1.16 the effect of a reduction from 1 m to about 10 cm is reported. Since in our first system the cables cannot be shorter than 1 m, measurements will be upper bounded to 10 MHz.
This is not applicable with the HP 4284A, because of its limitation to 1 MHz.
Figure 1.16: 50 Ω resistor measurement results. Black line refers to 1 m long cables. Grey line refers to 10 cm long cables.
1.4 Measurement results
The estimation of electrical parameters, by means of the cell constant, is not so direct when emulsions and polar solutions are measured. Moreover the dou- ble layer affects the measured values partially hiding the contribution of the solution in the low frequency range. Therefore, measurements were usually performed in a wide frequency range, analysing impedance and admittance in polar or linear planes, as real and imaginary parts, or as Cole-Cole plots. Sec- ondarily, experimental data were used for obtaining, by complex interpolation, an equivalent circuit model.
The complex interpolation is a common practice in electrochemistry for better separating different contributions. The most difficult step is to find an equivalent electrical circuit, that agrees with data, in which each element could describe a physical phenomenon occurring in the electrode-interface-electrolyte
(a) The complete equivalent cir- cuit.
(b) The simplified equivalent circuit.
Figure 1.17:Equivalent circuit of IDAM in air.
system. As an example, let considering the circuit reported in Fig. 1.17(a), showing the electrical model of the sensor in air: both air cover and glass sub- strate can be modelled by a RC circuit, but it is not possible to separate the contribution of the superstrate from that of the substrate until the conductiv- ity and the permittivity of one of them are known. For instance, with an air cover and referring to Fig. 1.17(b), experimental data point out for R1 = 41 GΩ and C1 = 16.7 pF that, considering the specified values of glass = 6.7 and air
= 1, allows obtaining Rglass = R1 (σglass σair) and Cglass= 14.5 pF.
Notice that the equivalent circuit should be carefully chosen, having in mind the physical phenomena, and not exceeding in complexity. Circuit model parameters were always extracted from measures in a wide range of frequen- cies, so that different physical phenomena could be separated. The use of a high number of measures provides an overdetermined system, and correlation coefficients allow evaluating the interpolation accuracy.
LEVM [32] is the Complex Nonlinear Least Square (CNLS) fitting program used for the numerical post-processing. This application needs an equivalent circuit with the initial value of each element and provides the interpolated results, together with several coefficients about the fit quality.
To speed up the post processing stage, several MATLAB scripts have been written in order to display measurement results in different ways and also to automatically interpolate them onto predefined circuits.
In the following section, the most relevant results will be shown together with the interpolations, when needed. Each measurement has been repeated
using at least two sensors and in different times. In the reported plots, only one set of measures is shown for clarity and, unless differently indicated, different repetitions gave always similar trends.
However, reproducibility has always been the main challenge because of the non-homogeneousness of milk.
1.4.1 DC measurements
DC measurements are important for deciding the signal level for the following measurements. As already explained in the first part of the chapter, the be- haviour of a metal-electrolyte interface is non-linear, schematizable by a pair of two antiparallel diodes, and non-linear phenomena should be avoided since difficult to unravel; therefore the signal level should be limited to the nearly linear area next to zero.
For this test, a direct connection between the interdigitated sensor and a Keithley 2400 SourceMeter was set up and the probe was dipped in full fat milk at about 23◦C.
Fig. 1.18(a) shows the I-V results. Notice that the capacitive nature of the electrode-electrolyte interface leads to long transient times: when a volt- age value is applied to the probe, the settling time is of several seconds (Fig. 1.18(b)). Thus, the current was always sampled with a certain delay with respect to the voltage steps. It should be also reminded that the DC current, established after a constant voltage application, is due to the faradaic resistance in parallel to the double layer capacitor.
The interpolation with the exponential Bulter Volmer equation (Eq. 1.12), gives the following values:
I = Is·(eηVTV + e−ηVTV ) = 18 × 10−9·(e0.32V + e−0.32V ) . (1.24) Observe that the interface capacitive behaviour is responsible for the non- zero current with null voltage, and that the direction of such a current depends on the polarity of the last potential applied. This can be explained considering the long discharge transients caused by the double layer.
After these first measurements it was decided to use always a signal level less than 300 mV. To confirm such a conclusion, some impedance measure- ments have been done on the full frequency range with different amplitudes of the excitation voltage. The results are shown in Fig. 1.19 and point out an alteration of the milk characteristics at low frequency for amplitudes higher
